## Approximable dimension and acyclic resolutions

### Volume 152 / 1997

Fundamenta Mathematicae 152 (1997), 43-53
DOI: 10.4064/fm-152-1-43-53

#### Abstract

We establish the following characterization of the approximable dimension of the metric space $X$ with respect to the commutative ring $R$ with identity: a-$\dim_R X \le n$ *if and only if there exist a metric space $Z$ of dimension at most n and a proper $UV^{n-1}$-mapping $f:Z \to X$ such that $\check H^n(f^{-1}(x);R) = 0 $ for all* $x \in X$. As an application we obtain some fundamental results about the approximable dimension of metric spaces with respect to a commutative ring with identity, such as the subset theorem and the existence of a universal space. We also show that approximable dimension (with arbitrary coefficient group) is preserved under refinable mappings.